summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=2 x^{4}-12 x^{2} )\non what interval(s) is ( f ) concave upward or concave downward?\n(type your answer in interval notation. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.)\na. ( f ) is concave upward on ( (-infty,-1),(1, infty) ) and ( f ) is concave downward on ( (-1,1) )\nb. ( f ) is never concave upward; ( f ) is concave downward on\nc. ( f ) is never concave downward; ( f ) is concave upward on\nsketch a graph of ( y = f(x) ). choose the correct graph below.\n

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=2 x^{4}-12 x^{2} )\non what interval(s) is ( f ) concave upward or concave downward?\n(type your answer in interval notation. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.)\na. ( f ) is concave upward on ( (-infty,-1),(1, infty) ) and ( f ) is concave downward on ( (-1,1) )\nb. ( f ) is never concave upward; ( f ) is concave downward on\nc. ( f ) is never concave downward; ( f ) is concave upward on\nsketch a graph of ( y = f(x) ). choose the correct graph below.\n

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (y = f(x)=2x^{4}-12x^{2}), the first derivative (y^\prime=f^\prime(x)=8x^{3}-24x).

Step2: Find the second derivative

Differentiate (y^\prime = 8x^{3}-24x) again. Using the power rule, (y^{\prime\prime}=f^{\prime\prime}(x)=24x^{2}-24 = 24(x^{2}-1)=24(x + 1)(x - 1)).

Step3: Determine concavity

Set (y^{\prime\prime}=0), then (24(x + 1)(x - 1)=0), which gives (x=-1) and (x = 1).

  • For (x\in(-\infty,-1)), let (x=-2). Then (y^{\prime\prime}(-2)=24((-2)^{2}-1)=24\times3>0), so (f(x)) is concave upward on ((-\infty,-1)).
  • For (x\in(-1,1)), let (x = 0). Then (y^{\prime\prime}(0)=24(0^{2}-1)=-24<0), so (f(x)) is concave downward on ((-1,1)).
  • For (x\in(1,\infty)), let (x = 2). Then (y^{\prime\prime}(2)=24(2^{2}-1)=24\times3>0), so (f(x)) is concave upward on ((1,\infty)).

Answer:

A. (f) is concave upward on ((-\infty,-1),(1,\infty)) and (f) is concave downward on ((-1,1))